Per ESPN's Tim MacMahon, Cousins "was not happy" playing behind Wood in head coach Stephen Silas' rotation and wanted more playing time.
Houston officially waived Cousins on Tuesday.
The Rockets acquired Wood in a sign-and-trade deal with the Detroit Pistons on Nov. 24. He was coming off a breakout season in 2019-20 with 13.1 points per game on 56.7 percent shooting and a 38.6 percent success from three-point range.
Wood, who has missed the past eight games with an ankle injury, has been Houston's primary starting center this season. The 25-year-old has started all 17 games he's played thus far. He leads the team with 22 points and 10.2 rebounds per game.
Cousins signed a one-year deal with the Rockets on Dec. 1 in an attempt to restart his career. The four-time All-Star appeared in just 78 games combined over the past three seasons because of injuries, including a torn Achilles in January 2018 and a torn ACL in August 2019.
ESPN's Adrian Wojnarowski reported on Feb. 19 that Houston was guaranteeing Cousins' $2.3 million contract for this season. Shams Charania of The Athletic and Stadium reported the following day that the Rockets and Cousins were planning to part ways.
Per Marc Berman of Fox 26, the Rockets guaranteed Cousins' deal because "management appreciates his professionalism and how he went about his business."
The Rockets used Cousins as their starting center in 11 of 25 games he played for them. He is averaging a career-low 9.6 points on 37.6 percent shooting and 7.6 rebounds per contest.